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Register a new userThe derivative map for diffeomorphism of disks: An example
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We prove that the derivative map $d colon mathrm{Diff}_partial(D^k) to Omega^kSO_k$, defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for $k = 11$ we prove that the following homomorphism is non-zero: $$ d_* colon pi_5mathrm{Diff}_partial(D^{11}) to pi_{5}Omega^{11}SO_{11} cong pi_{16}SO_{11} $$ As a consequence we give a counter-example to a conjecture of Burghelea and Lashof and so give an example of a non-trivial vector bundle $E$ over a sphere which is trivial as a topological $mathbb{R}^k$-bundle (the rank of $E$ is $k=11$ and the base sphere is $S^{17}$.) The proof relies on a recent result of Burklund and Senger which determines those homotopy 17-spheres bounding $8$-connected manifolds, the plumbing approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an explicit construction of low-codimension embeddings of certain homotopy spheres.
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The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.
A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N --> N where N is a manifold. The action of this operad on EC(j,M) (self embeddings R^j x M --> R^j x M with support in I^j x M) is an extension of the action of the operad of (j+1)-cubes on this space. Moreover the action of the splicing operad encodes Larry Siebenmanns splicing construction for knots in S^3 in the j=1, M=D^2 case. The space of long knots in R^3 (denoted K_{3,1}) was shown to be a free 2-cubes object with free generating subspace P, the subspace of long knots that are prime with respect to the connect-sum operation. One of the main results of this paper is that K_{3,1} is free with respect to the splicing operad action, but the free generating space is the much `smaller space of torus and hyperbolic knots TH subset K_{3,1}. Moreover, the splicing operad for K_{3,1} has a `simple homotopy-type as an operad.
We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago--Ivanov--Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which is the analog of the curve graph for the mapping class group.
The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.
For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contractible and the identity component H_0(M) of H(M) is an open normal subgroup in H_c(M). This induces the topological factorization H_c(M) approx H_0(M) times M_c(M) for the mapping class group M_c(M) = H_c(M)/H_0(M) with the discrete topology. Furthermore, for any non-compact surface M, the pair (H(M), H_c(M)) is locally homeomorphic to (square^w l_2,cbox^w l_2) at the identity id_M of M. Thus the group H_c(M) is an (l_2 times R^infty)-manifold. We also study topological properties of the group D(M) of diffeomorphisms of a non-compact smooth n-manifold M endowed with the Whitney C^infty-topology and the subgroup D_c(M) of D(M) consisting of all diffeomorphisms with compact support. It is shown that the pair (D(M),D_c(M)) is locally homeomorphic to (square^w l_2, cbox^w l_2) at the identity id_M of M. Hence the group D_c(M) is a topological (l_2 times R^infty)-manifold for any dimension n.
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